Real-rooted atlas

A038319 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(15 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 225 + 480\,x + 256\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0225480256961
133751080011520409629791
25062521600034560024576065536923521
37593754050000864000092160004915200104857628629151
411390625729000001944000002764800002211840009437184016777216887503681
517085937512757500004082400000725760000077414400004954521600176160768026843545627512614111
625628906252187000000081648000000174182400000232243200000198180864000105696460800322122547204294967296852891037441
7384433593753690562500001574640000000391910400000062705664000006688604160000475634073600021743271936005798205849606871947673626439622160671
85766503906256150937500000295245000000008398080000000015676416000000020065812480000017836277760000010871635968000043486543872000103079215104001099511627776819628286980801
986497558593751014904687500005412825000000001732104000000000369515520000000055180984320000005885971660800000448454983680000023917599129600008504035246080001814194185830401759218604441625408476896404831
1012974633789062516607531250000009743085000000000346420800000000008314099200000000014189395968000000017657914982400000016144379412480000010762919608320000051024211476480000163277476724736003166593487994880281474976710656787662783788549761
11194619506835937526987238281250000172718325000000000675520560000000000180138816000000000034586652672000000004918990602240000000524692330905600000041975386472448000002487430309478400000106130359871078400030874286507950080054887620458577920450359962737049624417546297445042591
122919292602539062543594769531250000030225706875000000001289630160000000000037829151360000000000807021895680000000001291235033088000000001574076992716800000001469138526535680000001044720729980928000005571843893231616000021612000555565056000576320014815068160094575592174780416072057594037927936756943935220796320321
13437893890380859375700630224609375000052313723437500000000241805655000000000000773778096000000000000181579926528000000000032280875827200000000004427091542016000000000472223097815040000000039177027374284800000002507329751954227200000121567503125053440000043224001111130112000010639754119662796800016212958658533785600115292150460684697623465261991844685929951
146568408355712890625112100835937500000000896806687500000000000446410440000000000000015475561920000000000000396174385152000000000007747410198528000000000011805577445376000000000014166692934451200000000013432123671183360000000010029319007816908800000058352401500025651200000259344006666780672000008511803295730237440000194555503902405427200027670116110564327424018446744073709551616727423121747185263828481
Technical details
Node id
n-a8ec826a617e4dc9 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d935ec63feee39a0

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           225,             480,             256;
  1:          3375,           10800,           11520,             4096;
  2:         50625,          216000,          345600,           245760,            65536;
  3:        759375,         4050000,         8640000,          9216000,          4915200,          1048576;
  4:      11390625,        72900000,       194400000,        276480000,        221184000,         94371840,         16777216;
  5:     170859375,      1275750000,      4082400000,       7257600000,       7741440000,       4954521600,       1761607680,        268435456;
  6:    2562890625,     21870000000,     81648000000,     174182400000,     232243200000,     198180864000,     105696460800,      32212254720,       4294967296;
  7:   38443359375,    369056250000,   1574640000000,    3919104000000,    6270566400000,    6688604160000,    4756340736000,    2174327193600,     579820584960,     68719476736;
  8:  576650390625,   6150937500000,  29524500000000,   83980800000000,  156764160000000,  200658124800000,  178362777600000,  108716359680000,   43486543872000,  10307921510400,   1099511627776;
  9: 8649755859375, 101490468750000, 541282500000000, 1732104000000000, 3695155200000000, 5518098432000000, 5885971660800000, 4484549836800000, 2391759912960000, 850403524608000, 181419418583040, 17592186044416;
%F Formula
The row polynomials satisfy P(n,x) = (15+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 480*x + 256*x^2.
%t Mathematica
P[0, x_] := 225 + 480*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(15 + 16*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 225 + 480*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (15 + 16*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.